What is Standard Deviation?

Standard deviation measures how spread out values are from the mean. A low standard deviation means data is clustered near the mean; a high value means data is widely spread.

Population σ = √[Σ(xᵢ − μ)² / N]
Sample s = √[Σ(xᵢ − x̄)² / (N−1)]

Steps to Calculate

Find the mean (average) of the dataset.
Subtract the mean from each value; square the result.
Find the average of all squared differences (variance).
Take the square root of the variance.

Standard Deviation Step-by-Step Worked Example

Dataset: {2, 4, 4, 4, 5, 5, 7, 9} (classic textbook example)

Step 1 – Mean: (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
Step 2 – Squared deviations:
(2−5)²=9, (4−5)²=1, (4−5)²=1, (4−5)²=1,
(5−5)²=0, (5−5)²=0, (7−5)²=4, (9−5)²=16
Step 3 – Sum = 9+1+1+1+0+0+4+16 = 32
Step 4 – Population variance = 32/8 = 4
Step 5 – Population SD σ = √4 = 2

Standard Deviation Reference Ranges

Range	% of Data (Normal Distribution)
Within ±1σ of mean	≈ 68.27%
Within ±2σ of mean	≈ 95.45%
Within ±3σ of mean	≈ 99.73% (the "3-sigma rule")

These ranges are the famous 68-95-99.7 empirical rule (bell curve rule), widely used in quality control and scientific research.

Coefficient of Variation (Relative SD)

CV = (σ / mean) × 100%
Allows comparison of variability across different datasets
Example: Dataset A has SD=10, mean=100 → CV=10%
Dataset B has SD=2, mean=10 → CV=20% (more variable despite smaller SD)

Real-World Applications

Finance: Portfolio risk — a stock with SD of 20% is twice as volatile as one with 10%
Quality control (Six Sigma): Manufacturing processes aim for ±6σ — defect rate under 3.4 per million
Weather forecasting: Temperature variability and climate change analysis
Academic testing: Standardized scores (Z-scores) use SD to normalize results

Standard Deviation Calculator

σ Standard Deviation Calculator